History of the calculus of variations in economics

Reginatto, Vinícius Oike

doi:10.11606/d.12.2019.tde-02102019-155737

Cited by 8 publications

(11 citation statements)

References 31 publications

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“…However the derivation by Reginatto, though enlightening, also creates a puzzle: Why is the inference method in statistical mechanics based on the information measure (2) while that used in obtaining Schrodinger's equation based on a different measure (9)? If, as discussed in [14], the use of the information measure (2) is well-motivated in statistical mechanics, then what is the motivation for using the Fisher information measure in the derivation of the Schrodinger equation? If there are several information measures, how does one decide a priori on which to choose?…”

Section: Physics Through Inferencementioning

confidence: 99%

“…However this is an artifact that is easy to remedy. Notice that in (13) ∆(x) can be either positive or negative but in (14) and the sequel it was implicitly assumed that L was positive. Therefore, in order to obtain a parity-invariant equation one should use a symmetrised form of Eq.…”

Section: Parity Preservationmentioning

confidence: 99%

“…Therefore, in order to obtain a parity-invariant equation one should use a symmetrised form of Eq. (14) involving both postive and negative values of L. Such a symmetrised measure is equivalent to the use in information theory of the J-divergence [4,5].…”

Section: Parity Preservationmentioning

confidence: 99%

“…For brevity, in latter constructions and discussions the focus will be on simple measures like (14) that violate parity maximally, as the more general case of parity conservation or small parity violation is easily obtained using a symmetrisation like (25) with the symbols having a meaning as highlighted following that equation. In order to avoid possible confusion with other forms of parity violation, the parity violation due to the length parameters L ± will sometimes be referred to as "geometric parity violation".…”

Section: Parity Preservationmentioning

confidence: 99%

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An information-theoretic link between spacetime symmetries and quantum linearity

Parwani

2005

Annals of Physics

105

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A nonlinear generalisation of Schrodinger's equation is obtained using information-theoretic arguments. The nonlinearities are controlled by an intrinsic length scale and involve derivatives to all orders thus making the equation mildly nonlocal. The nonlinear equation is homogeneous, separable, conserves probability, but is not invariant under spacetime symmetries. Spacetime symmetries are recovered when a dimensionless parameter is tuned to vanish, whereby linearity is simultaneously established and the length scale becomes hidden. It is thus suggested that if, in the search for a more basic foundation for Nature's Laws, an inference principle is given precedence over symmetry requirements, then the symmetries of spacetime and the linearity of quantum theory might both be emergent properties that are intrinsically linked. Supporting arguments are provided for this point of view and some testable phenomenological consequences highlighted. The generalised Klien-Gordon and Dirac equations are also studied, leading to the suggestion that nonlinear quantum dynamics with intrinsically broken spacetime symmetries might be relevant to understanding the problem of neutrino mass(lessness) and oscillations: Among other observations, this approach hints at the existence of a hidden discrete family symmetry in the Standard Model of particle physics.

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Section: Physics Through Inferencementioning

confidence: 99%

Section: Parity Preservationmentioning

confidence: 99%

Section: Parity Preservationmentioning

confidence: 99%

Section: Parity Preservationmentioning

confidence: 99%

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An information-theoretic link between spacetime symmetries and quantum linearity

Parwani

2005

Annals of Physics

105

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“…We recall first that the classical Fisher information associated with translations of a 1-D observable X with probability density P (x) (related to a quantum geometry probability measure ds 2 = [(dp j ) 2 /p j ]) is [40,43,85,96,97,98,99,160,161]). One has a well known Cramer-Rao inequality V ar(X) ≥ F −1 X where V ar(X) ∼ variance of X.…”

Section: Fisher Information Revisitedmentioning

confidence: 99%

Fluctuations, Information, Gravity and the Quantum Potential

Carroll¹

2006

144

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We show how the quantum potential arises in various ways and trace its connection to quantum fluctuations and Fisher information along with its realization in terms of Weyl curvature. It represents a genuine quantization factor for certain classical systems as well as an expression for quantum matter in gravity theories of Weyl-Dirac type. Many of the facts and examples are extracted from the literature (with references cited) and we mainly provide connections and interpretation, with a few new observations. We deliberately avoid ontological and epistemological discussion and resort to a collection of contexts where the quantum potential plays a visibly significant role. In particular we sketch some recent results of F. and A. Shojai on Dirac-Weyl action and Bohmian mechanics which connects quantum mass to the Weyl geometry. Connections a la Santamato of the quantum potential with Weyl curvature arising from a stochastic geometry, are also indicated for the Schrödinger equation (SE) and Klein-Gordon (KG) equation. Quantum fluctuations and quantum geometry are linked with the quantum potential via Fisher information. Derivations of SE and KG from Nottale's scale relativity are sketched along with a variety of approaches to the KG equation. Finally connections of geometry and mass generation via Weyl-Dirac geometry with many cosmological implications are indicated, following M. Israelit and N. Rosen.

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Quantum–classical correspondence via a deformed kinetic operator

Mosna¹,

Hamilton²,

Site³

2005

J. Phys. A: Math. Gen.

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We propose an approach to the quantum-classical correspondence based on a deformation of the momentum and kinetic operators of quantum mechanics. Making use of the factorization method, we construct classical versions of the momentum and kinetic operators which, in addition to the standard quantum expressions, contain terms that are functionals of the N-particle density. We show that this implementation of the quantum-classical correspondence is related to Witten's deformation of the exterior derivative and Laplacian, introduced in the context of supersymmetric quantum mechanics. The corresponding deformed action is also shown to be related to the Fisher information. Finally, we briefly consider the possible relevance of our approach to the construction of kinetic-energy density functionals. PACS numbers: 03.65.-w, 12.60.Jv, 89.70.+c, 31.15.Ew 1 Introduction Since the origins of quantum mechanics there has been interest in the correspondence between quantum and classical mechanics which has continued to the present [1]. In his 1926 paper [2] Schrödinger begins with the classical Hamilton-Jacobi equation and then writes down a wavefunction equation (now known as the Schrödinger equation) without making an explicit connection between the two. A general connection was made by Van Vleck in his 1928 paper [3] and extended by Schiller [4] who modifies the classical Hamilton-Jacobi equation to obtain a quantum-like formulation of classical mechanics. On the other hand, in his 1928 paper [5] Madelung begins with the wavefunction in polar form and then writes down hydrodynamic equations to obtain a classical-like formulation of quantum mechanics. This approach was extended by Bohm [6] who explicitly introduces the quantum potential, Q. One can think of the quantum-classical correspondence as "switching off" the quantum potential term in the modified (quantum) Hamilton-Jacobi equation [7] and this approach was explicitly explored in Ref. [8]. The Q → 0 and (more usual) → 0 approaches to the quantum-classical correspondence are discussed in Ref. [7] (see also Ref. [9]).In this paper we approach the quantum-classical correspondence at the level of the equations of motion of an N-particle system. Recall that, by expressing the wavefunction in polar form, the Schrödinger equation can be transformed into two equations [5, 10]: a modified Hamilton-Jacobi equation in which the quantum

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History of the calculus of variations in economics

Cited by 8 publications

References 31 publications

An information-theoretic link between spacetime symmetries and quantum linearity

An information-theoretic link between spacetime symmetries and quantum linearity

Fluctuations, Information, Gravity and the Quantum Potential

Quantum–classical correspondence via a deformed kinetic operator

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